Stone-von Neumann,
Uniqueness ofh 2 d+1representation
Uniqueness of Cliff(2d,C) representa-
tion on spinorsMp(2d,R) double cover ofSp(2d,R) Spin(n) double cover ofSO(n)
J:J^2 =− 1 , Ω(Ju,Jv) = Ω(u,v) J:J^2 =− 1 , (Ju,Jv) = (u,v)
M⊗C=M+J⊕M−J V ⊗C=V+J⊕VJ−
Coordinateszj∈M+J,zj∈M−J Coordinatesθj∈V+J,θj∈VJ−
U(d)⊂Sp(2d,R) commutes withJ U(d)⊂SO(2d,R) commutes withJ
CompatibleJ∈Sp(2d,R)/U(d) CompatibleJ∈O(2d)/U(d)
aj,a†jsatisfying CCR aFj,aF†jsatisfying CAR
aj| 0 〉= 0,| 0 〉depends onJ aFj| 0 〉= 0,| 0 〉depends onJ
H=Fdfin=C[z 1 ,...,zd] =S∗(Cd) H=Fd+= Λ∗(Cd)
a†j=zj, aj=∂z∂j a†j=θj, aj=∂θ∂j
Positivity conditions, leading to unitary state space:
Ω(v,Jv)>0 for non-zerov∈M (v,v)>0 for non-zerov∈V
〈u,u〉=iΩ(u,u)>0 for
non-zerou∈M+J
〈u,u〉= (u,u)>0 for
non-zerou∈VJ−orVJ+